General Mathematics

arXiv:math.GM

Mathematical material of general interest, broadly accessible to all mathematicians.

General Mathematics

arXiv:math.GM

Mathematical material of general interest, broadly accessible to all mathematicians.

Trending in General Mathematics

Analysis of the Riemann Zeta Function via Recursive Taylor Expansions

We present an unconditional proof that non-trivial zeros of the Riemann Zeta function must lie strictly on the critical line $\text{Re}(s) = 0.5$. By defining a recursive path of Taylor expansions originating from the domain of absolute convergence, we translate the zeta function towards the critical region, which is an easy-to-understand form of the analytical continuation. We then assume the existence of off-critical-line (off-line) zeros, which exist in pairs symmetric by the critical line. If the pairs are zero in value, their real and imaginary components differences should be both zero. However, we derive a contradiction against the assumption via basic logical deduction, proving the non-existence of the off-line zeros.

2603.05122

Mar 2026General Mathematics

Related categories:

Commutative Algebra Algebraic Geometry Analysis of PDEs Algebraic Topology Classical Analysis and ODEs

690 papers

Spectral Geometry of the Primes

We construct a family of self-adjoint operators on the prime numbers whose entries depend on pairwise arithmetic divergences, replacing geometric distance with number-theoretic dissimilarity. The resulting spectra encode how coherence propagates through the prime sequence and define an emergent arithmetic geometry. From these spectra we extract observables such as the heat trace, entropy, and eigenvalue growth, which reveal persistent spectral compression: eigenvalues grow sublinearly, entropy scales slowly, and the inferred dimension remains strictly below one. This rigidity appears across logarithmic, entropic, and fractal-type kernels, reflecting intrinsic arithmetic constraints. Analytically, we show that for the unnormalized Laplacian, the continuum limit of its squared Hamiltonian corresponds to the one-dimensional bi-Laplacian, whose heat trace follows a short-time scaling proportional to $t^{-1/4}$. Under the spectral dimension convention $d_s=-2\,d\logΘ/d\log t$, this result produces $d_s = 1/2$ directly from first principles, without fitting or external hypotheses. This value signifies maximal spectral compression and the absence of classical diffusion, indicating that arithmetic sparsity enforces a coherence-limited, non-Euclidean geometry linking spectral and number-theoretic structure.

2604.03351Apr 2026

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Exact Formulas for Coprime Representations of Even Integers Avoiding a Prime

Fix a prime $p \ge 5$ and define $g(2n,p)=\#\{(h,k)\in\mathbb{Z}_{>0}^2 : h+k=2n,\; h\le k,\; \gcd(h,6p)=\gcd(k,6p)=1\}$. We derive explicit closed-form expressions for $g(2n,p)$ in terms of the canonical remainder operator $δ_k(x)=x-k\lfloor x/k\rfloor$, elementary step functions, and the minimal solutions of the congruences $6x \equiv -1 \pmod{p}$ and $6x \equiv -5 \pmod{p}$. A key ingredient is an explicit formula for the minimal solution of $δ_k(a_0 x)=b_0$ obtained via the Euclidean algorithm, which determines the excluded residue classes directly. The resulting formulas show that $g(2n,p)$ is piecewise affine along arithmetic progressions of $n$, governed by residue classes modulo $3$ and $p$. For fixed $p$, after precomputing two residue parameters in $O(\log p)$ time, each evaluation of $g(2n,p)$ requires only $O(1)$ operations, compared to $O(n)$ for direct enumeration. The formulas are validated computationally for all $2n \le 10^5$ and primes $p \in \{5,7,11,13,17,19,23\}$, with perfect agreement with brute-force enumeration.

2604.02386Apr 2026

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Neural Prime Sieves: Density-Driven Generalization and Empirical Evidence for Hardy-Littlewood Asymptotics

Special prime families (twin, Sophie Germain, safe, cousin, sexy, Chen, and isolated primes) are central objects of analytic number theory, yet no efficiently computable probabilistic filter exists for identifying likely members among known primes at large scale. Classical sieves assign no probability weights to surviving candidates, and prior machine learning approaches are limited by the algorithmic randomness of the prime indicator sequence, yielding near-zero true positive rates. We present PrimeFamilyNet, a multi-head residual network conditioned on the backward prime gap and modular primorial residues of a known prime $p$, learning probabilistic filters for all seven families simultaneously and generalising across nine orders of magnitude from training ($10^7$--$10^9$) to evaluation at $10^{16}$. Isolated prime recall increased monotonically from $0.809$ at $5\times10^8$ to $0.984$ at $10^{16}$, a gain of $17.5$ percentage points and the only family among seven to improve with scale. Because recall is invariant to class prevalence, this reflects genuine decision boundary sharpening, not the rising isolated-prime fraction at extreme scales. A model trained only to $10^9$ reproduced the correct asymptotic direction without density supervision, corroborating Hardy--Littlewood $k$-tuple predictions. The causal model retained over $95\%$ recall for five families near $10^{10}$ while reducing the search space by $62$--$88\%$. For Chen primes, causal recall exceeded non-causal recall at every scale (margin $+0.245$ at $10^{16}$) because $g^+=2$ encodes only the prime case of the Chen condition. Focal Loss collapsed sparse algebraic family recall to $0.000$. Asymmetric Loss outperformed weighted BCE in-distribution but degraded more steeply out-of-distribution, showing that in-distribution recall alone is a misleading criterion for scale-generalisation tasks.

2604.02383Apr 2026

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2604.01258

RBF Kernel Parameter Formula for Data Classification Methods

Radial Basis Function (RBF), or Gaussian, kernels are among the most widely used parametric kernels in machine learning, particularly in methods such as Support Vector Machines (SVM) and kernel-based subspace approaches. The kernel parameter $γ$ (or $σ$ in the Gaussian formulation) must be carefully tuned, as the performance of these methods strongly depends on its value and is highly sensitive to improper selection. In practice, this parameter is typically determined through computationally expensive training procedures, which may also lack robustness. In this paper, we propose an efficient analytical formula for selecting the RBF kernel parameter that significantly reduces the computational cost of RBF-based methods. The proposed approach is derived by optimizing the diameter of mapped classes in the feature space while simultaneously maximizing inter-class feature distances. The detailed formulation is presented, and its efficiency is validated on the widely used SVM algorithm as well as on a Proper Orthogonal Decomposition (POD)-based subspace method for both binary and multi-class classification problems.

2604.01258Apr 2026

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A Theory of Scales and Orbit Covers

This paper develops a formal theory of musical scales and their harmonic coverings and introduces orbit covers: coverings obtained by translating a fixed subset across a scale via a group action. Orbit covers generalize familiar constructions, such as the covering of the diatonic scale by tertian triads, and are motivated by the search for a generalized harmonic framework extending common-practice tonality. We model modes as group structures associated with pitch-class sets and scales as torsors, introducing scale covers and, in particular, orbit covers. To each orbit cover we associate a nerve complex encoding its intersection structure and associated topological invariants. We classify triadic orbit covers of heptatonic scales up to affine symmetry and nerve isomorphism. These results support a broader theory of harmonic organization with analytical and compositional applications.

2604.02373Mar 2026

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2603.29988

Simplex Layers and Phase Boundaries in the Partition Graph

For the partition graph $G_n$ on the set of partitions of $n$, we study the stratification induced by the local simplex dimension $\dim_{\mathrm{loc}}(λ)$, defined as the maximal dimension of a simplex of the clique complex $K_n=\mathrm{Cl}(G_n)$ containing $λ$. Using the previously established description of maximal cliques through a vertex in terms of star and top capacities, we define the simplex layers $L_r(n):=\{λ\vdash n:\dim_{\mathrm{loc}}(λ)=r\}$ and study their global structure. We formalize the resulting layer stratification, rewrite layer membership in terms of local capacities, and record its basic consequences, including conjugation invariance. We then investigate first occurrence of layers across $n$, introducing the indices $n_r^{\mathrm{first}}$ and the corresponding first-occurrence sets $\mathcal{F}_r$. For the initial layer values, we obtain explicit exact results; more generally, we record a finite first-occurrence table and several natural sequence questions. We also define the adjacent-layer edge boundary $\partial^E_{r,r+1}(n)$, consisting of edges joining $L_r(n)$ to $L_{r+1}(n)$, together with the associated one-sided and vertex-boundary variants. This provides an exact interface language for the layer stratification, distinct from the broader shell-type geometric language used elsewhere in the project.

2603.29988Mar 2026

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2603.29831

On the shortest open cubic equations

We use cubic reciprocity to prove that the equation $7x^3+2y^3=3z^2+1$ has no integer solutions. Prior to this work, it was the shortest cubic equation for which the existence of integer solutions remained open. We conclude with a list of the new shortest open cubic equations.

2603.29831Mar 2026

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2604.00054

The Collision Spectrum

For a prime base $b$ and primitive odd Dirichlet character $χ$ modulo $b^2$, the collision transform coefficient $\hat{S}^{\circ}(χ)$ admits an exact factorization: \[ \hat{S}^{\circ}(χ) = -\frac{B_{1,\overlineχ} \cdot \overline{S_G(χ)}}{φ(b^2)}, \] where $B_{1,\overlineχ}$ is the generalized first Bernoulli number and $S_G(χ)$ is the diagonal character sum. By the standard Bernoulli--$L$-value formula, $|B_1| = (b/π)\, |L(1, χ)|$, so the collision invariant's Fourier spectrum encodes $L$-function special values. A Parseval identity gives an exact formula for the weighted second moment $\sum |L(1, χ)|^2 \cdot |S_G(χ)|^2$ in terms of the collision invariant's values on the finite group. The digit function computes this $L$-value moment exactly. Under a conditional zero-free hypothesis, the triangle inequality yields a separate bound connecting $L(1)$ to $L(s)$ for $s$ in the critical strip. At base~$5$, the factorization gives $|\hat{S}^{\circ}| \propto |L(1)|^2$ exactly. For quadratic characters in the family, the decomposition specializes to class-number data.

2604.00054Mar 2026

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Simplicial shells and thickness in the partition graph

For each positive integer $n$, let $G_n$ be the graph whose vertices are the partitions of $n$, with edges given by elementary transfers of one unit between parts, followed by reordering. We study the local simplex dimension in the clique complex $K_n=\Cl(G_n)$ as a geometric thickness invariant of $G_n$. For a partition $λ\vdash n$, let $τ_n(λ):=\dim_{\mathrm{loc}}(λ)$ be its simplicial thickness. This gives threshold thick zones $T_{\ge r}(n)=\{λ: τ_n(λ)\ge r\}$ and, relative to the boundary framework of $G_n$, a shell/core decomposition into outer shells $Sh_r(n)$ and inner cores $Core_r(n)$. Using local-morphology results established earlier in the series, we work with simplicial thickness as a local invariant. We prove that it is preserved by conjugation, that the induced thick zones, shells, and cores are conjugation-invariant, and that the antennas remain strictly one-dimensional in the simplicial sense and are excluded from all nontrivial thick zones. The first shell order at which a nontrivial shell can occur is therefore $2$, and the corresponding shell $Sh_2(n)$ is the triangular skin, while higher simplicial regimes form nested higher-order shells inside the triangular regime. We also develop a complete finite computational atlas for $1\le n\le 30$, giving first-occurrence tables for the regimes $T_{\ge r}(n)$ and supporting a finite-range rear-central thickening pattern.

2603.28171Mar 2026

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2604.00047

The Collision Transform

For a prime p and base b, the collision invariant $S_{\ell}(p)$, introduced in the companion paper, is a function of $p \bmod b^{\ell+1}$ and therefore lives on the finite group $(\mathbb{Z}/b^{\ell+1}\mathbb{Z})^{\times}$. Its Fourier expansion over Dirichlet characters modulo $b^{\ell+1}$ is the collision transform. The reflection identity forces all even-character coefficients of the centered invariant to vanish: only odd characters contribute. The centered prime harmonic sum $F^{\circ}(s) = \sum_p S^{\circ}_p / p^s$ is therefore a finite linear combination of non-trivial odd character sums $\sum_p χ(p)/p^s$, with no principal-character term. At $s = 1$, each sum converges by Mertens' theorem for arithmetic progressions. Convergence below $s = 1$ is conditional on the absence of $L$-function zeros above a given depth. Computation indicates convergence persists to at least $s = 0.6$ in base 10 and to $s = 0.5$ in base 3. The real parts of the products $\hat{S}^{\circ}(χ) \cdot P(s, χ)$ have mixed signs, so convergence is a collective constraint on the joint zero distribution, not a test of each $L$-function individually. Aggregating the collision deviation across bases with a fixed convergent weighting produces the base sum, a function on primes that reveals mod-3 structure. For bases with $3 \nmid b$, the reflection $a \mapsto m - a$ fixes a unique residue class modulo 3, and the mean of $S$ over units in that class equals the grand mean $-1/2$ (the neutrality theorem). Removing the mod-3 component introduces a principal-character term that is absent from $F^{\circ}$. The base-summed harmonic sum is negligible: the collision invariant's structural content is base-specific.

2604.00047Mar 2026

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2604.00045

The Collision Invariant

For a prime p and base b, the digit function delta(r) = floor(br/p) partitions the residues {1, ..., p-1} into b contiguous bins. The collision count C(g) records how many residues share a bin with their image under multiplication by g. We prove four results about this function. First, the gate width theorem: exactly b-1 multipliers satisfy C(g) = 0, given by the explicit family g = -u/(b-u) mod p for u = 1, ..., b-1. Second, the finite determination theorem: the collision deviation S at lag l depends only on p mod b^(l+1). Third, the reflection identity: S(a) + S(m-a) = -1 for m = b^(l+1), implying a grand mean of -1/2 and a pairing symmetry across the group of units. Fourth, the half-group theorem: for every non-trivial good slice n, the wrapping set W_n has size exactly phi(m)/2. The bilateral symmetry a -> m-a swaps wrapping with non-wrapping.

2604.00045Mar 2026

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2603.28822

Triangles Circumscribed about Central Conics and Their Invariants

We study families of triangles that are inscribed in a fixed circle and circumscribed about a central conic, extending the classical Chapple--Euler relation within the framework of Poncelet geometry. We establish several geometric invariants that arise when the circumcenter of the triangle coincides with either the center of the conic or one of its foci. These include invariance properties of orthic triangles, tangential triangles, polar circles, and trigonometric expressions such as $\sin^2 A + \sin^2 B + \sin^2 C$. We further derive explicit analytic constructions of central conics associated with a given triangle under special configurations, including cases where the foci are located at the circumcenter and orthocenter. In addition, we investigate extremal area problem within Poncelet families, and develop both homothetic and non-homothetic constructions of sequences of Poncelet pairs. These results provide a unified geometric framework linking classical triangle geometry, central conics, and Poncelet porism.

2603.28822Mar 2026

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2603.27248

Degree theory of the partition graph: exact maxima, profiles, and fibres

For the partition graph $G_n$, whose vertices are the partitions of $n$ and whose edges correspond to elementary unit transfers between parts, we develop a degree theory with three levels: exact value theory, exact profile theory, and fibre-level geometry. Writing $n=T_s+q$ with $T_s=s(s+1)/2$ and $0\le q\le s$, we prove that every degree-maximizing partition lies in the support-maximal stratum and obtain the exact formula \[ Δ_n=s(s-1)+\lfloor\sqrt{4q+1}\rfloor-1 \] for the maximal degree in $G_n$. For a support-maximal partition $λ$, let $A(λ)$ and $B(λ)$ denote the numbers of active gap bonuses and multiplicity bonuses. We prove that the set of realized maximizing profiles is \[ Π_n=\{(a,b)\in\mathbb Z_{\ge0}^2:a+b=ρ(q),\ T_a+T_b\le q\}, \qquad ρ(q)=\lfloor\sqrt{4q+1}\rfloor-1. \] Thus the exact global theory stops at the profile level. For each realized profile we then study the corresponding fibre of maximizers: we prove nonemptiness, construct canonical representatives, obtain lower bounds for mixed fibres, and show that conjugation induces a bijection between the fibres for $(a,b)$ and $(b,a)$. We also classify exactly the first near-triangular fibre windows and formulate localization and stability questions for the remaining fixed-$q$ regime.

2603.27248Mar 2026

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2603.28814

A Simple Trigonometric Classification of Quartic Roots

This article provides a simple trigonometric method for determining how many roots of a quartic equation are real and how many are complex, without solving the equation. The approach replaces the quartic's classical discriminant -- a degree-six polynomial in the coefficients -- with an elementary analysis of the function $f(θ) = a\cosθ+ \cos 4θ+ b$ on $[0,π]$, obtained by matching the quartic to the Chebyshev identity $8\cos^4\!θ- 8\cos^2\!θ+ 1 = \cos 4θ$. The derivation is computationally light and conceptually natural, and has the potential to demystify the geometry of a quartic equation's roots from a trigonometric perspective.

2603.28814Mar 2026

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Analytical continuation of Euler prime product for $\Re(s)>\tfrac{1}{2}$ assuming (RH)

We analytically continue the Euler prime product for $\Re(s)>\tfrac{1}{2}$ (except for its pole $s=1$) assuming (RH) by introducing a new factor to the Euler product. We also discuss how to recover the Mertens's 3rd Theorem at $s=1$ case, and how to apply the same technique to analytically continue other similar Euler products. In the last part, we also construct a simple script in Pari/GP to compute the Euler product and verify the calculations numerically.

2603.28808Mar 2026

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Numerical topology of the clique complex of the partition graph: Euler characteristic, clique counts, and sequence data

We study the numerical topology of the clique complex $K_n=\mathrm{Cl}(G_n)$, where $G_n$ is the partition graph on the set of integer partitions of $n$. Building on the previously established homotopy equivalence $K_n \simeq \vee^{\,b_n} S^2$, we shift the focus from qualitative topology to its numerical content. Our main objects are the Euler characteristic $χ(K_n)$, the derived sequence $b_n=χ(K_n)-1$, the clique counts $c_r(n)$, and several related maximal-simplex counts. We develop two exact counting languages for the same invariant. The first is the direct clique-counting formula $χ(K_n)=\sum_{r\ge 1}(-1)^{r-1}c_r(n)$, which expresses Euler characteristic through clique counts in the partition graph. The second is a nerve-side formula arising from the canonical good cover by distinct full star- and full top-simplices, which yields $χ(K_n)=χ(N_n)$, where $N_n$ is the corresponding nerve. We further use the classification of maximal simplices into star-, top-, and edge-type pieces to formulate a local-to-global counting framework based on local admissibility data and global deduplication. The paper is primarily organizational and computational. It fixes a consistent counting dictionary, separates intrinsic global counts from auxiliary based counts, records exact data for the full main sequence package on $1\le n\le 25$, and extends the low-dimensional clique-count layer through $n=60$. We do not claim closed formulas for $χ(K_n)$ or for the full family of clique counts. Rather, the paper provides a framework in which such questions can be studied systematically.

2603.26656Mar 2026

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2603.26854

A general framework for level continuous fuzzy-valued functions

In this paper we provide a general setting to deal with level continuous fuzzy-valued functions. Namely, we embed such functions into a product of spaces of real-valued functions of two variables satisfying certain types of left-continuity, right-continuity and monotonicity.

2603.26854Mar 2026

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2603.25917

Morphogenesis Across n: Overlays, Emergence Thresholds, and Weak Self-Similarity in the Partition Graph

We study the partition graphs $G_n$ as a growing family of discrete geometric objects and introduce a formal framework for comparing their structures across different levels. The main tool is a family of Ferrers-translation maps \[ T_τ:G_n\to G_{n+k},\qquad (T_τ(λ))'=λ'+τ', \] defined for fixed partitions $τ\vdash k$. We prove that these maps are induced graph embeddings, giving a rigorous notion of translation overlay: an induced copy of $G_n$ inside $G_{n+k}$. As a consequence, every finite rooted induced motif persists to all higher levels under translation overlays, and every overlay-monotone finitely witnessed property has a stable emergence threshold. We apply this framework to obtain monotonicity for the extremal local invariants $Δ_n$, $Ω_n$, and $S_n$, and to establish strict threshold statements for a canonical family of theorem-safe motifs drawn from boundary, axial, and rear morphology. This yields a conservative structural language for discussing growth across $n$ while keeping exact transport separate from stronger typed or visual interpretations. We also record a compact atlas framework for first appearances, repeated patterns, and comparative growth profiles. In this way the paper isolates a theorem-level core for persistence and thresholds, and complements it with a weaker notion of self-similarity based on recurring finite motifs and repeated local fragments.

2603.25917Mar 2026

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Directional Geometry and Anisotropy in the Partition Graph

We develop a directional formalism for the partition graph G_n based on several canonical reference sets: the main chain, the self-conjugate axis, the spine, and the boundary framework. For each such set S, the graph distance d_S induces a shell structure and a local trichotomy of edges into inward, outward, and level classes. Passing from edges to paths, we define directional corridors as monotone inward geodesics toward a chosen reference set and prove that every vertex admits at least one. We then prove a structural non-equivalence theorem: for connected G_n, two nonempty reference sets induce the same edgewise directional field if and only if the difference of their distance functions is constant; in particular, distinct reference sets induce distinct directional fields. This gives a first precise formalization of anisotropy in G_n. We also show that every bounded neighborhood of a reference set is accessible by a monotone inward corridor, which gives a directional interpretation to previously established controlled regions around the axis, the spine, and the framework. Finally, we complement the strict theory with a computational atlas illustrating edgewise directional statistics, directional mixing, local invariant drift, and corridor-based transport profiles.

2603.25488Mar 2026

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2603.24824

Boundary Framework, Rear Morphology, and Rectangular Ears in the Partition Graph

We study the outer geometry of the partition graph $G_n$, focusing on its canonical front-and-side framework, the family of nontrivial rectangular partitions, and the rear structures suggested by the visible geometry of the graph. We formalize the boundary framework $\mathcal B_n=\mathcal M_n\cup\mathcal L_n\cup\mathcal R_n$, where $\mathcal M_n$ is the main chain and $\mathcal L_n,\mathcal R_n$ are the left and right side edges, and we isolate the nontrivial rectangular family $\mathrm{Rect}^*(n)=\{(a^b):ab=n,\ a,b\ge2\}$ as a canonical discrete family marking the rear part of $G_n$. We prove that every nontrivial rectangular vertex $ρ=(a^b)$ has degree $2$, has exactly two explicitly described neighbors, and lies in a unique triangle of $G_n$. This leads to the notions of a rectangular ear, its attachment pair, and its support edge. We also prove that $\mathrm{Rect}^*(n)$ is an independent set in $G_n$, so the weak rectangular contour is not a graph-theoretic chain but a discrete rear marker family. For every genuinely rear rectangular ear, namely for $a,b\ge3$, we show that its support edge lies in a tetrahedral configuration of the clique complex $K_n=\mathrm{Cl}(G_n)$. To organize the interaction between different ears, we introduce support zones, support distances, and support corridors between attachment pairs. The paper also records a natural divisor-theoretic indexing of the rectangular family, presents a computational atlas in small and large ranges, and concludes with open problems concerning support-zone connectivity, inter-ear corridors, and canonical rear contours in $G_n$.

2603.24824Mar 2026

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General Mathematics Papers (math.GM) - ScienceStack

690 papers

Spectral Geometry of the Primes

2604.03351Apr 2026

View

Exact Formulas for Coprime Representations of Even Integers Avoiding a Prime

2604.02386Apr 2026

View

Neural Prime Sieves: Density-Driven Generalization and Empirical Evidence for Hardy-Littlewood Asymptotics

2604.02383Apr 2026

View

2604.01258

RBF Kernel Parameter Formula for Data Classification Methods

2604.01258Apr 2026

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A Theory of Scales and Orbit Covers

2604.02373Mar 2026

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2603.29988

Simplex Layers and Phase Boundaries in the Partition Graph

2603.29988Mar 2026

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2603.29831

On the shortest open cubic equations

2603.29831Mar 2026

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2604.00054

The Collision Spectrum

2604.00054Mar 2026

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Simplicial shells and thickness in the partition graph

2603.28171Mar 2026

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2604.00047

The Collision Transform

2604.00047Mar 2026

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2604.00045

The Collision Invariant

2604.00045Mar 2026

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2603.28822

Triangles Circumscribed about Central Conics and Their Invariants

2603.28822Mar 2026

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2603.27248

Degree theory of the partition graph: exact maxima, profiles, and fibres

2603.27248Mar 2026

View

2603.28814

A Simple Trigonometric Classification of Quartic Roots

2603.28814Mar 2026

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Analytical continuation of Euler prime product for $\Re(s)>\tfrac{1}{2}$ assuming (RH)

2603.28808Mar 2026

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Numerical topology of the clique complex of the partition graph: Euler characteristic, clique counts, and sequence data

2603.26656Mar 2026

View

2603.26854

A general framework for level continuous fuzzy-valued functions

2603.26854Mar 2026

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2603.25917

Morphogenesis Across n: Overlays, Emergence Thresholds, and Weak Self-Similarity in the Partition Graph

2603.25917Mar 2026

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Directional Geometry and Anisotropy in the Partition Graph

2603.25488Mar 2026

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2603.24824

Boundary Framework, Rear Morphology, and Rectangular Ears in the Partition Graph

2603.24824Mar 2026

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